THE GENERALIZED PASCAL MATRIX VIA THE GENERALIZED FIBONACCI MATRIX AND THE GENERALIZED PELL MATRIX

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1 J Korean Mah Soc , No, pp THE GENERALIZED PASCAL MATRIX VIA THE GENERALIZED FIBONACCI MATRIX AND THE GENERALIZED PELL MATRIX Gwang-yeon Lee and Seong-Hoon Cho Reprined from he Journal of he Korean Mahemaical Sociey Vol 45, No, March 008 c 008 The Korean Mahemaical Sociey

2 J Korean Mah Soc , No, pp THE GENERALIZED PASCAL MATRIX VIA THE GENERALIZED FIBONACCI MATRIX AND THE GENERALIZED PELL MATRIX Gwang-yeon Lee and Seong-Hoon Cho Absrac In [4], he auhors sudied he Pascal marix and he Sirling marices of he firs kind and he second kind via he Fibonacci marix In his paper, we consider generalizaions of Pascal marix, Fibonacci marix and Pell marix And, by using Riordan mehod, we have facorizaions of hem We, also, consider some combinaorial ideniies Inroducion The Pascal numbers, he Fibonacci number and he Pell number are very ineresing in combinaorial analysis For inegers n, i and j, n i, j, he n n Pascal marix P n [p ij ] is defined by { i p ij j if i j, 0 oherwise In [], he auhors gave marices represenaions of he Pascal riangle More generally, for nonzero real variable x, he Pascal marix was generalized in P [x] n and Q[x] n, respecively which are defined in [8], and hese generalized Pascal marices were also exended in Φ[x, y] n [ϕ[x, y] ij ] see [9] for any wo nonzero real variables x and y where { i ϕ[x, y] ij j x i j y i+j if i j, 0 oherwise In his paper, we call Φ[x, y] n he GP marix By he definiion, we have P [x] n Φ[x, ] n, Q[y] n Φ[, y] n, P n P [] n Q[] n Φ[, ] n, [ ] i P [x] n P [ x] n x i j i j j Received Augus 9, 006; Revised Ocober 30, Mahemaics Subjec Classificaion 05A9, 05B0, B39 Key words and phrases Pascal marix, Fibonacci marix, Pell marix, Riordan marix This paper was suppored by Hanseo Universiy, c 008 The Korean Mahemaical Sociey

3 480 GWANG-YEON LEE AND SEONG-HOON CHO In [8] and [9], he facorizaions of P [x] n, Q[x] n, and Φ[x, y] n are obained, respecively The Fibonacci sequence has been discussed in so many aricles and books In [5], he auhors inroduced he Fibonacci marix F n [f ij ] of order n as follows { Fi j+ i j + 0, F n [f ij ] 0 i j + < 0, where F k is he kh Fibonacci number Now, we inroduce a generalizaion of he Fibonacci marix For any wo nonzero real numbers x and y and posiive ineger n, he n n GF marix F[x, y] n [f[x, y] ij ] defined by { Fi j+ x f[x, y] ij i j y i+j if i j, 0 oherwise By he definiion, we have F[, ] n F n In [5], he auhors gave he Cholesky facorizaion of he Fibonacci marix F n and hey also discussed eigenvalues of he symmeric Fibonacci marix In [4], he auhors sudied he Pascal marix and he Sirling marix of he firs kind and of he second kind via he Fibonacci marix and some combinaorial ideniies are obained from he marices represenaions of he Pascal marix, he Sirling marices and he Fibonacci marix The Pell sequence {a n } is defined recursively by he equaion a n+ a n + a n for n, where a, a The Pell sequence is,, 5,, 9, 70, 69, 408, In [] and [3], he auhors gave well-known Pell ideniies as follows, for arbirary inegers q and r a n+q a n+r a n a n+q+r a q a r n, a n+ a n + a n+, and for nh Pell number a n, a n [n /] r0 n r r n r We define n n Pell marix S n [s ij ] as follows { ai j+, i j + 0, s ij 0 oherwise, where a n is he nh Pell number Now, we consider a generalizaion of he Pell marix For any wo nonzero real numbers x and y and posiivie ineger n, he n n generalized Pell marix,

4 G PASCAL VIA G FIBONACCI AND PELL 48 S[x, y] n [s[x, y] ij ] defined by s[x, y] ij { ai j+ x i j y i+j, i j + 0, 0 oherwise, where a n is he nh Pell number By he definiion, we have S[, ] n S n In [7], he auhors inroduced a group which called he Riordan group, and hey gave some applicaions in he group In [6], he auhors inroduced Riordan marix, and hey proved ha each Riordan marix R in he group can be facorized by Pascal marix P, Caalan marix C and Fibonacci marix F as R P CF In [7], he Riordan group was defined as follows: Le R [r ij ] i,j 0 be an infinie marix wih enries in he complex numbers Le c i n 0 r n,i n be he generaing funcion of he ih column of R We call R a Riordan marix if c i g[f] i, where g + g + g + g 3 3 +, and f + f + f In his case, we wrie R g, f We denoe by R he se of Riordan marices Then he se R is a group under marix muliplicaion,, wih he following properies: R g, f h, l ghf, lf R I, is he ideniy elemen R3 The inverse of R is given by R, f, where f is he g f composiional inverse of f, ie, f f ff R4 If a 0, a, a, T is a column vecor wih generaing funcion A, hen muliplying R g, f on he righ by his column vecor yields a column vecor wih generaing funcion B gaf We call R a Riordan group From he definiion of he Riordan marix, we know ha he marices in he Riordan group are infinie and lower riangular Here are hree examples abou he Riordan marices The firs example of elemen in R is he Pascal marix, and he following represenaion is well-known P g P, f P,

5 48 GWANG-YEON LEE AND SEONG-HOON CHO We consider he infinie Fi- The nex example is he Fibonacci marix bonacci marix F [F ij ] as follows; F g F, f F Since he firs column of F is,,, 3, 5, T, i is obvious ha g F n0 F n+ n The rule of formaion in F is ha each enry is he sum of he elemens in he upper wo rows In oher words, F n+,j F n,j + F n,j, j So, we have f F because c j g F [f F ] j j, ie, F g F, f F, and hence F is in R Finally, we consider he infinie Pell marix S [s ij ] as follows; S g S, f S Since he firs column of S is,, 5,, 9, T, we have g S By he rule of formaion in S, i is obvious ha f S Tha is, S g S, f S, and hence S is also in R In his paper, we consider he relaionships beween GP marix and GF marix and generalized Pell marix S[x, y] Also, we give some ineresing combinaorial ideniies Main heorems To begin wih, we define a marix For any wo nonzero real variables x and y, an infinie marix L[x, y] [l[x, y] ij ] is defined as follows: i i i 3 l[x, y] ij x i j y j i j j j

6 G PASCAL VIA G FIBONACCI AND PELL 483 From he definiion of L[x, y], we see ha l[x, y], l[x, y] j 0 for j, l[x, y] 0, l[x, y] and l[x, y] j 0 for j 3 Also, we see ha l[x, y] i x i y i for i 3, and, for i, j, l[x, y] ij l[x, y] i,j + l[x, y] i,j xy From, we know ha l ij saisfy he Pascal-like recurrence relaion Using he definiions of Φ[x, y], F[x, y] and L[x, y], we can derive he following heorem Theorem Le L[x, y] be he infinie marix as in For he infinie GP marix Φ[x, y] and he infinie GF marix F[x, y], we have Φ[x, y] F[x, y] L[x, y] Proof From he definiions of he GP marix and GF marix, we have he following Riordan represenaions 3 Φ[x, y] xy, y, F[x, y] xy xy xy, y We know ha if L[x, y] is in R, hen we may assume L[x, y] g L, f L From, we have he infinie marix L[x, y] [l[x, y] ij ] as follows: L[x, y] x y xy x 3 y 3 0 xy 0 0 x 4 y 4 x 3 y 3 x y 3xy 0 Since he firs column vecor of L[x, y] is, 0, x y, x 3 y 3, T, i is obvious ha g L x y + x 3 y xy xy xy The rule of formaion in L[x, y] is ha each enry is he sum of he elemens o he lef and above in he row above In oher words, for j, Thus ie, l[x, y] n+,j l[x, y] n,j + l[x, y] n,j xy c j c j + xy c j, g L [f L ] j g L [f L ] j + xy g L [f L ] j or f L + xy f L Solving for f L, we have f L xy Thus, xy xy L[x, y] xy, xy

7 484 GWANG-YEON LEE AND SEONG-HOON CHO Therefore, F[x, y] L[x, y] xy xy, y xy xy, xy xy xy xy, y xy Φ[x, y], he proof is compleed xy xy xy, y xy y xy Since x and y are nonzero real variables, from Theorem, we have, for j n, n n k k k 3 4 F n j+ j j j j kj By 4, we have F + F + + F n F n+, and his ideniy is he sum of he firs n erms of he Fibonacci sequence From 4, we have, for k j +, n n k 3!jk j k j F n k+ j j!k j! kj Now, we consider he relaionship beween GP marix and generalized Pell marix For any wo nonzero real variables x and y, we define an infinie marix M[x, y] [m[x, y] ij ], i, j as follows: 5 m[x, y] ij i j i j i 3 x i j y j i j From he definiion of M[x, y], we have m[x, y], m[x, y] j 0 for j, m[x, y] xy, m[x, y] and m[x, y] j 0 for j 3 Also, we have m[x, y] i x i y i for i 3, and, for i, j, 6 m[x, y] ij m[x, y] i,j + m[x, y] i,j xy From 6, we know ha m[x, y] ij saisfy he Pascal-like recurrence relaion From he definiion of Φ[x, y], S[x, y] and M[x, y], we can derive he following heorem Theorem Le M[x, y] be he infiniie marix as in 5 For he infinie GP marix Φ[x, y] and he infinie generalized Pell marix S[x, y], we have Φ[x, y] S[x, y] M[x, y]

8 G PASCAL VIA G FIBONACCI AND PELL 485 Proof From he definiion of S[x, y], xy y S[x, y] 5x y xy 3 y x 3 y 3 5x y 4 xy 5 y 6 0 0, 9x 4 y 4 x 3 y 5 5x y 6 xy 7 y 8 0 and we know ha S[x, y] is in R So, we have he following represenaion S[x, y] xy xy, y We know ha if M[x, y] is in R, hen we may assume M[x, y] g M, f M From 5, we have xy M[x, y] x y x 3 y 3 x y xy 0 0 x 4 y 4 4x 3 y 3 x y xy 0 Since he firs column vecor of M[x, y] is i is obvious ha, xy, x y, x 3 y 3, x 4 y 4, T, g M xy x y x 3 y 3 3 x 4 y 4 4 xy xy xy 3 xy 4 xy xy xy The rule of formaion in M[x, y] is ha each enry is he sum of he elemens o he lef and above in he row above Tha is, for j m[x, y] ij m[x, y] i,j + m[x, y] i j,j xy Thus c j c j + xy c j, and hence g M [f M ] j g M [f M ] j + xy g M [f M ] j Thus, we have f M xy and xy xy M[x, y] xy, xy

9 486 GWANG-YEON LEE AND SEONG-HOON CHO Therefore, we have xy xy y S[x, y] M[x, y], xy xy xy xy xy, y xy Φ[x, y] The proof is compleed By Theorem and Theorem, we know ha for posiive ineger n, F[x, y] n Φ[x, y] n L[x, y] n and S[x, y] n Φ[x, y] n M[x, y] n Thus we consider inverse marices Le L[x, y] n be an n n marix as in From he definiion of L[x, y] n, he inverse marix L[x, y] n of L[x, y] n is of he form L[x, y] n [ l[x, y] ij ] wih [ ] i i i 3 l[x, y] ij i+j + x i j y j i, j j j and hence we have, for j, l[x, y] ij l[x, y] i,j l[x, y] i,j xy For he marix F[x, y], f F y because f F y So, we have Thus, g F f F xy xy F[x, y] xy xy, y Also, for he marix L[x, y], we see ha f L +xy and xy +xy g L f L xy xy Hence, we have he following lemma +xy + xy + xy xy +xy Lemma 3 Le F[x, y] be he infinie GF marix and le L[x, y] be he marix as in Then we have F[x, y] xy xy, y L[x, y] + xy + xy xy, + xy

10 G PASCAL VIA G FIBONACCI AND PELL 487 From he definiion of S[x, y] and M[x, y], we consider he inverse marices For he marix S[x, y], f S y because f S y Thus we have and hence, we have g S f S xy xy, S[x, y] xy xy, y +xy and xy xy For he marix M[x, y], we have f M g M f M + xy xy Hence, we have he following lemma +xy +xy xy +xy Lemma 4 Le S[x, y] be he infinie generalized Pell marix and le M[x, y] be he marix as in 5 Then we have S[x, y] xy xy, y + xy M[x, y] xy, + xy From Theorem, Theorem, Lemma 3 and Lemma 4, we have he following corollaries Corollary 5 For he infinie GP marix Φ[x, y], Φ[x, y] L[x, y] F[x, y] M[x, y] S[x, y] + xy, y + xy Proof From Theorem and Theorem 4, we know ha Thus, we have Φ[x, y] L[x, y] F[x, y] M[x, y] S[x, y] Φ[x, y] L[x, y] F[x, y] + xy + xy xy, + xy xy xy, y + xy, y + xy

11 488 GWANG-YEON LEE AND SEONG-HOON CHO Corollary 6 Le u n xy + xy xy n For posiive ineger n, we have i Φ[x, y] n u, y n n u n, Φ[x, y] n y n y n +u, n y n +u n n ii F[x, y] n k, y n, xy k xy k F[x, y] n n k xy k+ xy k+, y n n iii S[x, y] n k, y n, xy k xy k S[x, y] n n k xy k+ xy k+, y n Proof i From 3, we have Φ[x, y] xy + xy 3, y 4 xy + xy 3 By inducion on n for n, we can ge he Φ[x, y] n as follows: Φ[x, y] n Φ[x, y] n Φ[x, y] u n, y n u n u n, y n u n Since Φ[x, y] n y n y n +u n Thus, u, y n n u n xy, y xy, we have f y n +u n, and g f y Φ[x, y] n n y n + u n, y n + u n The proofs of ii and iii are similar o i Therefore, he proof is compleed Corollary 7 Le S[x, y] be he infinie generalized Pell marix Then we have xy F[x, y] S[x, y] xy xy, Proof From Theorem, Theorem and Lemma 3, we have F[x, y] S[x, y] M[x, y] L[x, y] xy xy + xy T S[x, y] xy + xy T xy T, T + xy T xy S[x, y] xy xy,, where T xy

12 G PASCAL VIA G FIBONACCI AND PELL 489 Corollary 8 For Φ[x, y], F[x, y], S[x, y], and posiive ineger n, he following resuls hold: i Φ[ x, y] n Φ[x, y] n, Φ[x, y] n Φ[ x, y] n ii F[ x, y] n F[x, y] n, F[x, y] n F[ x, y] n iii S[ x, y] n S[x, y] n, S[x, y] n S[ x, y] n Example If x y, hen, from Theorem, we have Φ[, ] P,, F[, ] F,, and L[, ] L, Then, we have F L,, P, From i of Corollary 6, we have P n n, 7 n, Le P n [p n ij ] for n From 7 and he definiion of he Pascal marix, we have he ineresing resul as follows: p n ij n i j i j n i j p ij From he Corollary 5, he nh column of P has n + n + + n+ as is generaing funcion And, we have, for n, P n + n, + n If x and y, hen, from 3, we have Φ[, ] +, P + More generally, for posiive ineger n, we have, Φ[, ] n P n

13 490 GWANG-YEON LEE AND SEONG-HOON CHO Also, from ii of Corollary 6 and by inducion on k for k, we have k F k,, k, Example If x y, hen, from Theorem 4, we have Φ[, ] P,, S[, ] S,, M[, ], So, we have P,,, and from iii of Corollary 6 and by inducion on k for k, we have S k k,, k, We consider he 7 7 marices P 7 S 7 M 7 Tha is, From his muliplicaion, we can ge many ideniies For example, 6 5 p 6,3 0 3 a 4 + a 3 + a + a 5 More generally, we have i p ij s i m j + s i m j + j a i m j + a i m j + + a m i,j + a m ij,

14 G PASCAL VIA G FIBONACCI AND PELL 49 where a n is he nh Pell number For example, for j, i i a i a i 3 4a i 4 i 4a i 3a, where a n is he nh Pell number In paricular, from he above ideniy, we have, for j, a n a n a n + + a, ie, a n a n + a n + + a + Also, we have, for posiive ineger n and Pell sequence {a n }, a + a + + a n a n+ a n+ a n+ + a n The above ideniy is he sum of he firs n erms of he Pell sequence From Corollary 7, we have he following ideniy F n a n a n F + a n F + + a F n, where a n is he nh Pell number References [] R Brawer and M Pirovino, The linear algebra of he Pascal marix, Linear Algebra Appl 74 99, 3 3 [] J Ercolano, Marix generaors of Pell sequences, Fibonacci Quar 7 979, no, 7 77 [3] A F Horadam, Pell ideniies, Fibonacci Quar 9 97, no 3, 45 5, 63 [4] G Y Lee, J S Kim, and S H Cho, Some combinaorial ideniies via Fibonacci numbers, Discree Appl Mah , no 3, [5] G Y Lee, J S Kim, and S G Lee, Facorizaions and eigenvalues of Fibonacci and symmeric Fibonacci marices, Fibonacci Quar 40 00, no 3, 03 [6] P Pear and L Woodson, Triple facorizaion of some Riordan marices, Fibonacci Quar 3 993, no, 8 [7] L W Shapiro, S Geu, W J Woan, and L C Woodson, The Riordan group, Discree Appl Mah 34 99, no -3, 9 39 [8] Z Zhizheng, The linear algebra of he generalized Pascal marix, Linear Algebra Appl , 5 60 [9] Z Zhizheng and M Liu, An exension of he generalized Pascal marix and is algebraic properies, Linear Algebra Appl 7 998, Gwang-yeon Lee Deparmen of Mahemaics Hanseo Universiy Chung-Nam , Korea address: gylee@hanseoackr Seong-Hoon Cho Deparmen of Mahemaics Hanseo Universiy Chung-Nam , Korea address: shcho@hanseoackr